A Finite Element - Capacitane Method for Elliptic Problems on Regions Partitioned into Subregions.
A Finite Element Merthod Scheme for one Dimensional Elliptic Equations with High Super Convergence at the Nodes.
A finite volume method for the Laplace equation on almost arbitrary two-dimensional grids
We present a finite volume method based on the integration of the Laplace equation on both the cells of a primal almost arbitrary two-dimensional mesh and those of a dual mesh obtained by joining the centers of the cells of the primal mesh. The key ingredient is the definition of discrete gradient and divergence operators verifying a discrete Green formula. This method generalizes an existing finite volume method that requires “Voronoi-type” meshes. We show the equivalence of this finite volume...
A finite volume method for the Laplace equation on almost arbitrary two-dimensional grids
We present a finite volume method based on the integration of the Laplace equation on both the cells of a primal almost arbitrary two-dimensional mesh and those of a dual mesh obtained by joining the centers of the cells of the primal mesh. The key ingredient is the definition of discrete gradient and divergence operators verifying a discrete Green formula. This method generalizes an existing finite volume method that requires “Voronoi-type” meshes. We show the equivalence of this finite volume...
A finite-dimensional reduction method for slightly supercritical elliptic problems.
A Fortin operator for two-dimensional Taylor-Hood elements
A standard method for proving the inf-sup condition implying stability of finite element approximations for the stationary Stokes equations is to construct a Fortin operator. In this paper, we show how this can be done for two-dimensional triangular and rectangular Taylor-Hood methods, which use continuous piecewise polynomial approximations for both velocity and pressure.
A free boundary problem for semilinear elliptic equations.
A free boundary value problem in potential theory
This paper is devoted to the formulation and solution of a free boundary problem for the Poisson equation in the plane. The object is to seek a domain and a function defined in satisfying the given differential equation together with both Dirichlet and Neumann type data on the boundary of . The Neumann data are given in a manner which permits reformulation of the problem as a variational inequality. Under suitable hypotheses about the given data, it is shown that there exists a unique solution...
A Galerkin procedure for approximating the flux on the boundary for elliptic and parabolic boundary value problems
A Generalization of the Schwarz Alternating Method to an Arbitrary Number of Subdomains.
A Harnack inequality approach to the regularity of free boundaries. Part III : existence theory, compactness, and dependence on
A hierarchical preconditioner for the mortar finite element method.
A Littlewood-Paley type inequality with applications to the elliptic Dirichlet problem
Let L be a strictly elliptic second order operator on a bounded domain Ω ⊂ ℝⁿ. Let u be a solution to in Ω, u = 0 on ∂Ω. Sufficient conditions on two measures, μ and ν defined on Ω, are established which imply that the norm of |∇u| is dominated by the norms of and . If we replace |∇u| by a local Hölder norm of u, the conditions on μ and ν can be significantly weaker.
A matching of singularities in domain decomposition methods for reaction-diffusion problems with discontinuous coefficients
In this paper we certify that the same approach proposed in previous works by Chniti et al. [C. R. Acad. Sci. 342 (2006) 883–886; CALCOLO 45 (2008) 111–147; J. Sci. Comput. 38 (2009) 207–228] can be applied to more general operators with strong heterogeneity in the coefficients. We consider here the case of reaction-diffusion problems with piecewise constant coefficients. The problem reduces to determining the coefficients of some transmission conditions to obtain fast convergence of domain decomposition...
A matching of singularities in domain decomposition methods for reaction-diffusion problems with discontinuous coefficients
In this paper we certify that the same approach proposed in previous works by Chniti et al. [C. R. Acad. Sci.342 (2006) 883–886; CALCOLO45 (2008) 111–147; J. Sci. Comput.38 (2009) 207–228] can be applied to more general operators with strong heterogeneity in the coefficients. We consider here the case of reaction-diffusion problems with piecewise constant coefficients. The problem reduces to determining the coefficients of some transmission conditions to obtain fast convergence of domain decomposition...
A mathematical and computational framework for reliable real-time solution of parametrized partial differential equations
We present in this article two components: these components can in fact serve various goals independently, though we consider them here as an ensemble. The first component is a technique for the rapid and reliable evaluation prediction of linear functional outputs of elliptic (and parabolic) partial differential equations with affine parameter dependence. The essential features are (i) (provably) rapidly convergent global reduced–basis approximations — Galerkin projection onto a space spanned...
A Mathematical and Computational Framework for Reliable Real-Time Solution of Parametrized Partial Differential Equations
We present in this article two components: these components can in fact serve various goals independently, though we consider them here as an ensemble. The first component is a technique for the rapid and reliable evaluation prediction of linear functional outputs of elliptic (and parabolic) partial differential equations with affine parameter dependence. The essential features are (i) (provably) rapidly convergent global reduced–basis approximations — Galerkin projection onto a space WN spanned...
A method for solving a boundary value problem for a partial differential equation of elliptic type.
A minimax formula for the principal eigenvalues of Dirichlet problems and its applications.
A mixed boundary value problem for the Laplace equation in an angle (2nd part)